Name for matrices with orthogonal (not necessarily orthonormal) rows Is there a name for a matrix whose rows (or columns) are non-zero orthogonal vectors ?
It seems to me that "orthogonal matrix" would be a good name, but this is already taken -- it refers to a matrix whose rows (or columns) form an orthonormal set of vectors.
 A: An orthogonal matrix refers to a matrix whose rows and columns are orthonormal. This is a key property of orthogonal matrices, one which ultimately requires these matrices to be square.
Suppose $A$ is rectangular matrix ($n > m$) with row and column vectors which are [a] non-zero and [b] orthogonal to one another.  We know,


*

*Orthogonal vectors are also linearly independent.

*The row rank of $A$ equals the column rank of $A$, $\textrm{rank}(A') = \textrm{rank}(A)$.


Then, (1) and (2) together suggest $\textrm{rank}(A') = \textrm{rank}(A)$, or $m = n$.  But this a contradiction.
Restricting the rows or columns to be orthogonal and non-zero is a departure of sorts.  A semi-orthogonal matrix $B$ is a non-square matrix with real entries having the property that either (1) $BB' = I_m$ or (2) $B'B = I_n$, with the respective true case representing an orthonormal basis.
The case you speak of, a matrix whose rows or columns are orthogonal (not orthonormal), could be described as a semi-orthogonal matrix under a scaling transformation.
A: Geometrically, two non-orthogonal rows mean a shear transformation, don't they? If this is right (and I'm sure somebody will let me know if I'm wrong), then you could call them shear-free matrices. Updated to add: Somebody let me know I was wrong, as predicted...
But a pure shear is volume-preserving, so this might be misleading. How about row-orthogonal matrices?
